The Markström graph is a cubic planar graph on 24 vertices which lacks cycles of length 4 and 8 but contains cycles of length 16. (In particular, it contains cycles of lengths 3, 5, 6, 7, and 9-24.)
The Markström graph is implemented in the Wolfram Language as GraphData["MarkstroemGraph"].
Markström (2004) found four cubic graphs on 24 vertices in which the only power-of-two graph cycle length is 16. The Markström graph is the only planar graph among the four. It is therefore not a counterexample to the Erdős-Gyárfás conjecture, but is a small extremal example related to the conjecture.
The Markström graph is a 
cyclic group graph.
